MA THEM A TIC A L . 1 .Y. 1 LYSIS OF RA NDOM XOISE 135 



{inx). This is a function of u. The ch. f. of two random variables x and 

 V is the average value of exp {iux -\- ivy) and is a function of u and v. The 

 ch. f. which we shall use here is the ch. f. of the two random variables V{t) 

 and Vit + t) where T'(/) is the voltage applied to the non-linear device, and 

 the randomness is introduced by / being selected at random, t remaining 

 lixed. We may write this characteristic function as 



1 r 

 g{u, V, t) = Limit - / exp [/«F(/) + ivV{t + r)] dt (4.8-1) 



r—w 1 Jo 



If T'(0 contains a noise voltage T'iv(Oj as it always does in this section, and 

 if we use the representation (2.8-1) or (2.8-6) a large number of random 

 parameters (dnS and 6„'s or (^,,'s) will appear in (4.8-1). In accordance 

 with our use of such representations we may average over these parameters 

 without changing the value of (4.8-1) and may thereby simpHfy the integra- 

 tion. 



For example suppose 



V{t) = VM + F^(/) (4.8-2) 



where !'.,(/) is some regular voltage which may, e.g., consist of one or more 

 sine waves. Substituting this in (4.8-1) and using the result (3.2-7) that 

 the ch. f. of Fjv(/) and Vx{t -f t) is 



gx(u, T, -) = ave. exp [iuVM{i) + ^vV^'(t -f- r)] 



— ^ («' + V') - ^rUV 



= exp -— {u -]- V-) - ip 



\p^ = \1/(t) being the correlation function of Fv(0> we obtain for the ch. f. 



of T'(/) and F(/ + t), 



g{u, V, t) = exp 



■y (//" + V') — XprtlV 



X Limit ]- [ exp [iuVs(t) + ivV,{t + r)] dt ^^'^'^^ 



= A'.v(", V, T)g,{u, V, t) 



In the last line we have used gs{u, v, r) to denote the limit in the line above: 



1 r'' 



g^{ti, V, t) = Limit - / exp [iuVsiO + ivVs{t + r)] dt (4.8-5) 



7'_oo I Jq 



The principal reason we use the ch. f. is because quite a few non-linear 

 devices may be described by the integral 



I = ^ f Fiiiije'"'' du (4.\-l) 



27r J c 



